My children receive social security benefits from their father. Do I pay taxes on it or does he?

[Hal_Al]

Neither.

Social security payments to children.

 If the Social Security (SS) payments. Including SS disability, were under the child's SS number (as they usually are), it does not get reported on your return. If it does need to be reported, it would go on the child's individual return. If that was his/her only income, it does not get reported at all. She does not need to file a tax return. 

SS is only taxable & reportable when added to sufficient other income. Social security only  becomes taxable, when his income, including 1/2 his social security, reaches $25,000